Proj construction

Proj of a graded ringEdit

Proj as a setEdit

Let S be a graded ring, where   is the direct sum decomposition associated with the gradation. The irrelevant ideal of S is the ideal of elements of positive degree  .

We say an ideal is homogeneous if it is generated by homogeneous elements. Then, as a set,


For brevity we will sometimes write X for  .

Proj as a topological spaceEdit

We may define a topology, called the Zariski topology, on   by defining the closed sets to be those of the form


where a is a homogeneous ideal of S. As in the case of affine schemes it is quickly verified that the V(a) form the closed sets of a topology on X.

Indeed, if   are a family of ideals, then we have   and if the indexing set I is finite, then  .

Equivalently, we may take the open sets as a starting point and define


A common shorthand is to denote D(Sf) by D(f), where Sf is the ideal generated by f. For any ideal a, the sets D(a) and V(a) are complementary, and hence the same proof as before shows that the sets D(a) form a topology on  . The advantage of this approach is that the sets D(f), where f ranges over all homogeneous elements of the ring S, form a base for this topology, which is an indispensable tool for the analysis of  , just as the analogous fact for the spectrum of a ring is likewise indispensable.

Proj as a schemeEdit

We also construct a sheaf on  , called the “structure sheaf” as in the affine case, which makes it into a scheme. As in the case of the Spec construction there are many ways to proceed: the most direct one, which is also highly suggestive of the construction of regular functions on a projective variety in classical algebraic geometry, is the following. For any open set U of   (which is by definition a set of homogeneous prime ideals of S not containing  ) we define the ring   to be the set of all functions


(where   denotes the subring of the ring of fractions   consisting of fractions of homogeneous elements of the same degree) such that for each prime ideal p of U:

  1. f(p) is an element of  ;
  2. There exists an open subset V of U containing p and homogeneous elements s, t of S of the same degree such that for each prime ideal q of V:
    • t is not in q;
    • f(q) = s/t.

It follows immediately from the definition that the   form a sheaf of rings   on  , and it may be shown that the pair ( ,  ) is in fact a scheme (this is accomplished by showing that each of the open subsets D(f) is in fact an affine scheme).

The sheaf associated to a graded moduleEdit

The essential property of S for the above construction was the ability to form localizations   for each prime ideal p of S. This property is also possessed by any graded module M over S, and therefore with the appropriate minor modifications the preceding section constructs for any such M a sheaf, denoted  , of  -modules on  . This sheaf is quasicoherent by construction. If S is generated by finitely many elements of degree 1 (e.g. a polynomial ring or a homogenous quotient of it), all quasicoherent sheaves on   arise from graded modules by this construction.[1] The corresponding graded module is not unique.

The twisting sheaf of SerreEdit

For related information, and the classical Serre twist sheaf, see tautological bundle

A special case of the sheaf associated to a graded module is when we take M to be S itself with a different grading: namely, we let the degree d elements of M be the degree (d + 1) elements of S, and denote M = S(1). We then obtain   as a quasicoherent sheaf on  , denoted   or simply O(1), called the twisting sheaf of Serre. It can be checked that O(1) is in fact an invertible sheaf.

One reason for the utility of O(1) is that it recovers the algebraic information of S that was lost when, in the construction of  , we passed to fractions of degree zero. In the case Spec A for a ring A, the global sections of the structure sheaf form A itself, whereas the global sections of   here form only the degree-zero elements of S. If we define


then each O(n) contains the degree-n information about S, and taken together they contain all the grading information that was lost. Likewise, for any sheaf of graded  -modules N we define


and expect this “twisted” sheaf to contain grading information about N. In particular, if N is the sheaf associated to a graded S-module M we likewise expect it to contain lost grading information about M. This suggests, though erroneously, that S can in fact be reconstructed from these sheaves; however, this is true in the case that S is a polynomial ring, below. This situation is to be contrasted with the fact that the spec functor is adjoint to the global sections functor in the category of locally ringed spaces.

Projective n-spaceEdit

If A is a ring, we define projective n-space over A to be the scheme


The grading on the polynomial ring   is defined by letting each   have degree one and every element of A, degree zero. Comparing this to the definition of O(1), above, we see that the sections of O(1) are in fact linear homogeneous polynomials, generated by the   themselves. This suggests another interpretation of O(1), namely as the sheaf of “coordinates” for  , since the   are literally the coordinates for projective n-space.

Examples of ProjEdit

  • If we let the base ring be  , then   has a canonical projective morphism to the affine line   whose fibers are elliptic curves, except at the points   where the curves degenerate into nodal curves.
  • The projective hypersurface  is an example of a Fermat quintic threefold which is also a Calabi–Yau manifold.
  • Weighted projective spaces can be constructed using a polynomial ring whose variables have non-standard degrees. For example, the weighted projective space   corresponds to taking   of the ring   where   have weight   while   has weight 2.
  • Having a bigraded ring corresponds to taking a subscheme of a product of projective spaces. For example, the bigraded algebra  , where the   have weight   and the   have weight  , corresponds to the ring of  .

Global ProjEdit

A generalization of the Proj construction replaces the ring S with a sheaf of algebras and produces, as the end result, a scheme which might be thought of as a fibration of Proj's of rings. This construction is often used, for example, to construct projective space bundles over a base scheme.


Formally, let X be any scheme and S be a sheaf of graded  -algebras (the definition of which is similar to the definition of  -modules on a locally ringed space): that is, a sheaf with a direct sum decomposition


where each   is an  -module such that for every open subset U of X, S(U) is an  -algebra and the resulting direct sum decomposition


is a grading of this algebra as a ring. Here we assume that  . We make the additional assumption that S is a quasi-coherent sheaf; this is a “consistency” assumption on the sections over different open sets that is necessary for the construction to proceed.


In this setup we may construct a scheme   and a “projection” map p onto X such that for every open affine U of X,


This definition suggests that we construct   by first defining schemes   for each open affine U, by setting


and maps  , and then showing that these data can be glued together “over” each intersection of two open affines U and V to form a scheme Y which we define to be  . It is not hard to show that defining each   to be the map corresponding to the inclusion of   into S(U) as the elements of degree zero yields the necessary consistency of the  , while the consistency of the   themselves follows from the quasi-coherence assumption on S.

The twisting sheafEdit

If S has the additional property that   is a coherent sheaf and locally generates S over   (that is, when we pass to the stalk of the sheaf S at a point x of X, which is a graded algebra whose degree-zero elements form the ring   then the degree-one elements form a finitely-generated module over   and also generate the stalk as an algebra over it) then we may make a further construction. Over each open affine U, Proj S(U) bears an invertible sheaf O(1), and the assumption we have just made ensures that these sheaves may be glued just like the   above; the resulting sheaf on   is also denoted O(1) and serves much the same purpose for   as the twisting sheaf on the Proj of a ring does.

Proj of a quasi-coherent sheafEdit

Let   be a quasi-coherent sheaf on a scheme  . The sheaf of symmetric algebras   is naturally a quasi-coherent sheaf of graded  -modules, generated by elements of degree 1. The resulting scheme is denoted by  . If   is of finite type, then its canonical morphism   is a projective morphism.[2]

For any  , the fiber of the above morphism over   is the projective space   associated to the dual of the vector space   over  .

If   is a quasi-coherent sheaf of graded  -modules, generated by   and such that   is of finite type, then   is a closed subscheme of   and is then projective over  . In fact, every closed subscheme of a projective   is of this form.[3]

Projective space bundlesEdit

As a special case, when   is locally free of rank  , we get a projective bundle   over   of relative dimension  . Indeed, if we take an open cover of X by open affines   such that when restricted to each of these,   is free over A, then


and hence   is a projective space bundle.

Example of Global ProjEdit

Global proj can be used to construct Lefschetz pencils. For example, let   and take homogeneous polynomials   of degree k. We can consider the ideal sheaf   of   and construct global proj of this quotient sheaf of algebras  . This can be described explicitly as the projective morphism  .

See alsoEdit


  1. ^ Ravi Vakil (2015). Foundations of Algebraic Geometry (PDF)., Corollary 15.4.3.
  2. ^ EGA, II.5.5.
  3. ^ EGA, II.5.5.1.